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Motivated by recent progress of quantum technologies, we study a discretizedquantum adiabatic process for a one-dimensional free fermion system describedby a variational wave function, i.e., a parametrized quantum circuit. The wavefunction is composed of $M$ layers of two elementary sets of time-evolutionoperators, each set being decomposed into commutable local operators. Theevolution time of each time-evolution operator is treated as a variationalparameter so as to minimize the expectation value of the energy. We show thatthe exact ground state is reached by applying the layers of time-evolutionoperators as many as a quarter of the system size. This is the minimum number$M_B$ of layers set by the limit of speed, i.e., the Lieb-Robinson bound, forpropagating quantum entanglement via the local time-evolution operators.Quantities such as the energy $E$ and the entanglement entropy $S$ of theoptimized variational wave function with $M < M_B$ are independent of thesystem size $L$ but fall into some universal functions of $M$. The developmentof the entanglement in these ansatz is further manifested in the progressivepropagation of single-particle orbitals in the variational wave function. Wealso find that the optimized variational parameters show a systematic structurethat provides the optimum scheduling function in the quantum adiabatic process.We also investigate the imaginary-time evolution of this variational wavefunction, where the causality relation is absent due to the non-unitarity ofthe imaginary-time evolution operators, thus the norm of the wave functionbeing no longer conserved. We find that the convergence to the exact groundstate is exponentially fast, despite that the system is at the critical point,suggesting that implementation of the non-unitary imaginary-time evolution in aquantum circuit is highly promising to further shallow the circuit depth.

Discretized quantum adiabatic process for free fermions and comparison with the imaginary-time evolution